Craps and Betting Systems

Date: 03/17/2003 at 11:10:28
From: Cliff Thomas
Subject: The game of Craps
In craps, can I bet on the 7 and win (more than I lose) if I assume a
seven normally comes up within 32 rolls?
I plan to only start betting after I have not seen a seven for 5 rolls
so I start betting on roll #6. (I know that the past doesn't influence
future dice rolls.) Then I gradually increase my bets to ensure a
profit of about $10 when I finally hit. The confusing part is whether
over the long term I can win my bankroll amount before losing it all
to when that stretch of no 7 extends beyond 32 rolls. And the fact
that I am not betting for 5 rolls after a 7, which means when a 7
comes up sometimes and I am neither winning or losing.
Not being a gambler I found that if I hop the 7 in bets of multiples
of $3, I could increase my returns and allow myself to extend my rolls
to 34 rolls still using my $6k bankroll. I am still trying to find
math that can clue me in as to whether I am likely to win my bankroll
back before losing it all to a streak of no 7 for 34 or more rolls. Is
there a formula to help me figure this out?

Date: 04/11/2003 at 13:20:31
From: Doctor Achilles
Subject: Re: The game of Craps
Hi Cliff,
Thanks for writing to Dr. Math.
There is a house edge on every bet in the casino. Craps is actually
one of the games with the smallest house edge, if you stick to the
pass line and lay odds on every bet. Laying odds on your pass and
come bets is actually the only place in the casino where you can make
a bet where there is no house edge. In other words, the payoff on
laying odds is exactly equal to the probability that the outcome will
occur.
My personal favorite site for gambling rules and advice is
The Wizard of Odds
http://www.wizardofodds.com/
The author of the site is extremely methodical and mathematical and
he does an outstanding job of explaining in mathematically sound ways
how each game works and the pros and cons of different strategies.
You can go directly to the craps page at:
http://wizardofodds.com/games/craps/
The problem with doubling your bet (or multiplying it by 3, or by
1.5, or using some other strategy of increasing your bet every time
you lose) is something that you hinted at already: you stand to lose
your entire bankroll for the potential of a small profit.
You can put a couple of thousand dollars on the line at a $10 craps
table and double your bet on the "don't pass" line every time you
lose. If you start with, say, $1000 and you double your bet on the
don't pass line until you either win once (and come out $10 ahead) or
lose all your money, then you have a little less than a 99% of coming
out ahead $10. If you start with $10000 and do the same thing, then
you have a little less than a 99.9% chance of winning.
If you turn that around, you have just over a 1% chance of losing in
the $1000 case and just over a 0.1% chance of losing in the $10000
case.
If you lose, then you are down either $1000 or $10000. A little more
than 1% multiplied by $1000 is equal to a little more than $10 and a
little more than 0.1% multiplied by $10000 is equal to a little more
than $10.
So, on most trials you will win $10, but you have a small chance of
losing a whole lot of money. And, on average, you will lose a bit
more than you win.
If you do this only once, then you will probably win $10, but you may
be unlucky and lose it all.
If you keep trying this over and over, then your odds of winning every
time drop significantly.
Let's say that you start with $1000 and play a $10 table, doubling
your bet until you win, then you start over at $10 and repeat. If you
keep doing this until you are either up to a total of $2000 or down to
nothing, then the odds that you will win are EXACTLY the same as if
you just sat down at a $1000 craps table and laid all your money down
on the first shooter.
In other words, on average, you expect to lose just as much as anyone
else, using any betting strategy (assuming you put the same amount of
money on the table). You can find a more complete explanation of
betting strategies and the problems with them back at the Wizard of
Odds:
http://wizardofodds.com/gambling/scam.html
It sounds like a great idea to sit at home with an Excel spreadsheet
and plan out betting strategies with which you increase your bet. I've
done it more than once. I once took one of my betting strategies
(which involved multiplying my bet) to a casino. At one point while I
was doing the strategy, I ran into a run of bad luck and had to double
my bet 6 times in a row on a $5. All of a sudden, I had over $300
riding on a $5 table. At that point, things didn't look quite as good
as they did on my Excel spreadsheet. Because the dice don't remember
how many times you've lost, you're no more likely to win after
doubling your bet 6 or 7 times than you are the first time. So here I
was with over $300 bet, and slightly less than a 50% chance of ending
$5 ahead of where I started and slightly more than a 50% chance of
ending several hundred dollars lower than where I started.
Fortunately, I won that bet (and walked away with my $5 profit), but
the experience taught me that betting systems are the sort of thing
that look nice on paper, and are fine before you start using them. If
you're lucky and conservative, they can be fun and (with enough luck)
even profitable. But increasing your bet when you're down will
occasionally leave you in a position where you either have to give up
or bet more money than you want to lose so that you can have a few
more dollars when you leave the casino than when you entered it.
Hope this helps. If you have other questions or you'd like to talk
about this some more, please write back.
- Doctor Achilles, The Math Forum
http://mathforum.org/dr.math/